Fix $\varepsilon>0$, a finite time interval $[0,T]$, and initial data $g_0$ for a three-dimensional Ricci flow with surgery. There exist constants $r>0$, $\bar\delta>0$, and $\theta>0$, depending only on $\varepsilon,T,g_0$ and the chosen standard-cap model, with the following property. Suppose that immediately before a surgery time $T_i\le T$ the flow satisfies Hamilton-Ivey pinching, is $\kappa$-noncollapsed on all balls with radii between the surgery scale and $r$, and has the $\varepsilon$-canonical-neighbourhood property at every point with scalar curvature at least $r^{-2}$; that is, each such point is modelled after rescaling on an $\varepsilon$-neck, an $\varepsilon$-cap, or a compact positively curved component. Suppose the surgery at $T_i$ is performed along pairwise disjoint strong $\delta$-necks of radius $h$ with $0<\delta\le\bar\delta$ and $0<h\le\bar\delta r$, and that each inserted cap is $\delta$-close in the prescribed high-$C^k$ topology to the normalized standard cap after scaling by $h^{-2}$. Then the post-surgery metric at $T_i^+$ has the $\varepsilon$-canonical-neighbourhood property at every point with scalar curvature at least $r^{-2}$. Moreover the restarted smooth flow preserves this conclusion on the time interval $[T_i,T_i+\theta h^2]\cap[0,T]$, as long as no later surgery has occurred.